# Parametrization of subgrid scales in long-term simulations of the shallow-water equations using machine learning and convex limiting

arXiv: https://arxiv.org/abs/2602.00378 · HTML: https://mlsanity.com/paper/2602.00378/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 62 references that are in the map.
- First posted on arXiv 30 Jan 2026.

## Builds on (most cited first, 8 of 8)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Deep learning to represent sub-grid processes in climate models](https://mlsanity.com/paper/1806.04731/) (2018)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains](https://mlsanity.com/paper/2007.11207/) (2020)

## Built on by (most cited first, 2 of 2)

- [Subgrid-Scale Parameterization in Burgers' Equation Using Structure-Preserving Neural Networks and Entropy Variables](https://mlsanity.com/paper/2607.14855/) (2026)
- [Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problem](https://mlsanity.com/paper/2607.27433/) (2026)

## Abstract (arXiv)

We present a method for parametrizing sub-grid processes in the Shallow Water equations. We define coarse variables and local spatial averages and use a feed-forward neural network to learn sub-grid fluxes. Our method results in a local parametrization that uses a four-point computational stencil, which has several advantages over globally coupled parametrizations. We demonstrate numerically that our method improves energy balance in long-term turbulent simulations and also accurately reproduces individual solutions. The long-term simulations refer to numerical studies where a fluid flow is simulated over a duration long enough to reach a statistical steady state. The neural network parametrization can be easily combined with flux limiting to reduce oscillations near shocks. More importantly, our method provides reliable parametrizations, even in dynamical regimes that are not included in the training data.